140 Prof. F. L. 0. Wadsworth on the 



I£ therefore we diaphragm off the left-hand half of the 

 grating-surface, the spectral image at the point +Kq will be 

 practically free from aberration ; if we diaphragm off the 

 right-hand half, the spectral image at the point — acq will 

 likewise be sharply defined. With gratings of the usual 

 angular apertures we can therefore always secure sharp 

 definition with the A-B-F types of O.S. mounting on any 

 given spectral line Xo by suitably choosing the angle i such 

 that 



p sin jS {sin [tang ~^( + sin /9 cos i)'] — sin i\ =mn\Q, . (36) 



and then covering the right- or the left-hand half of the 

 ruled surface. 



For gratings of large angular aperture, such as /^g, this 

 device gives less satisfactory results. The aberration of the 

 centre and extreme elements, a, h, or a, b' (fig. 1) is of 

 course eliminated at the point k^, defined by (34) and (35), 

 but the aberration for the intermediate points (which cor- 

 respond to different values of yS) remains. In the case of 

 grating ySg for example, for which Kq= +37^ at 2=30^, the 

 aberration due to the element intermediate between elements 

 a and h will be 



Ip sin4y82[-6-46 tang 37' + '035 (1 - 9*95 tang^ 37')sec2 37^] 

 = i/)sin^^i(2-5)^(-3-4)-53 K ; 



I. e. the aberration of the intermediate element of the grating 

 ySs is nearly as great as that of the extreme elements of the 

 grating /3i at the limit of the 5° field (see Table II. S_^^) and 

 over thirty times as great as that of the entire surface of ^i 

 at the centre of the field Sq". 



Examination of the formulae (16);, (16a), (26), {2Qa), and 

 (33), or of Table III., shows that for very small values of /c, 

 I. e., very near the axis of the grating, the symmetrical 

 aberration, which is expressed by the second term of these 

 equations, is of the greater numerical value ; for larger values, 

 the unsymmetrical aberration expressed by the first term 

 becomes by far the most important. Since the first varies 

 in absolute amount as sin^/3 and the second only as sin^/5 it at 

 once follows that the relative differences over a given field, 

 + /C, become smaller as the value of /3 increases, i. e. while 

 the absolute amount of aberration is greater for all parts of 

 the field, the definition, or rather lack of definition, is more 

 uniform. 



In considering the effect of a given aberration on the 

 performance of an optical instrument, it is usually assumed 



